Metamath Proof Explorer

Theorem hdmaprnlem7N

Description: Part of proof of part 12 in Baer p. 49 line 19, s-St e. G(u'+s) = P*. (Contributed by NM, 27-May-2015) (New usage is discouraged.)

Ref Expression
Hypotheses hdmaprnlem1.h 
|- H = ( LHyp ` K )
hdmaprnlem1.u 
|- U = ( ( DVecH ` K ) ` W )
hdmaprnlem1.v 
|- V = ( Base ` U )
hdmaprnlem1.n 
|- N = ( LSpan ` U )
hdmaprnlem1.c 
|- C = ( ( LCDual ` K ) ` W )
hdmaprnlem1.l 
|- L = ( LSpan ` C )
hdmaprnlem1.m 
|- M = ( ( mapd ` K ) ` W )
hdmaprnlem1.s 
|- S = ( ( HDMap ` K ) ` W )
hdmaprnlem1.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
hdmaprnlem1.se 
|- ( ph -> s e. ( D \ { Q } ) )
hdmaprnlem1.ve 
|- ( ph -> v e. V )
hdmaprnlem1.e 
|- ( ph -> ( M ` ( N ` { v } ) ) = ( L ` { s } ) )
hdmaprnlem1.ue 
|- ( ph -> u e. V )
hdmaprnlem1.un 
|- ( ph -> -. u e. ( N ` { v } ) )
hdmaprnlem1.d 
|- D = ( Base ` C )
hdmaprnlem1.q 
|- Q = ( 0g ` C )
hdmaprnlem1.o 
|- .0. = ( 0g ` U )
hdmaprnlem1.a 
|- .+b = ( +g ` C )
hdmaprnlem1.t2 
|- ( ph -> t e. ( ( N ` { v } ) \ { .0. } ) )
hdmaprnlem1.p 
|- .+ = ( +g ` U )
hdmaprnlem1.pt 
|- ( ph -> ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) )
Assertion hdmaprnlem7N 
|- ( ph -> ( s ( -g ` C ) ( S ` t ) ) e. ( L ` { ( ( S ` u ) .+b s ) } ) )

Proof

Step Hyp Ref Expression
1 hdmaprnlem1.h 
 |-  H = ( LHyp ` K )
2 hdmaprnlem1.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 hdmaprnlem1.v 
 |-  V = ( Base ` U )
4 hdmaprnlem1.n 
 |-  N = ( LSpan ` U )
5 hdmaprnlem1.c 
 |-  C = ( ( LCDual ` K ) ` W )
6 hdmaprnlem1.l 
 |-  L = ( LSpan ` C )
7 hdmaprnlem1.m 
 |-  M = ( ( mapd ` K ) ` W )
8 hdmaprnlem1.s 
 |-  S = ( ( HDMap ` K ) ` W )
9 hdmaprnlem1.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
10 hdmaprnlem1.se 
 |-  ( ph -> s e. ( D \ { Q } ) )
11 hdmaprnlem1.ve 
 |-  ( ph -> v e. V )
12 hdmaprnlem1.e 
 |-  ( ph -> ( M ` ( N ` { v } ) ) = ( L ` { s } ) )
13 hdmaprnlem1.ue 
 |-  ( ph -> u e. V )
14 hdmaprnlem1.un 
 |-  ( ph -> -. u e. ( N ` { v } ) )
15 hdmaprnlem1.d 
 |-  D = ( Base ` C )
16 hdmaprnlem1.q 
 |-  Q = ( 0g ` C )
17 hdmaprnlem1.o 
 |-  .0. = ( 0g ` U )
18 hdmaprnlem1.a 
 |-  .+b = ( +g ` C )
19 hdmaprnlem1.t2 
 |-  ( ph -> t e. ( ( N ` { v } ) \ { .0. } ) )
20 hdmaprnlem1.p 
 |-  .+ = ( +g ` U )
21 hdmaprnlem1.pt 
 |-  ( ph -> ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) )
22 eqid 
 |-  ( -g ` C ) = ( -g ` C )
23 1 5 9 lcdlmod 
 |-  ( ph -> C e. LMod )
24 lmodabl 
 |-  ( C e. LMod -> C e. Abel )
25 23 24 syl 
 |-  ( ph -> C e. Abel )
26 1 2 3 5 15 8 9 13 hdmapcl 
 |-  ( ph -> ( S ` u ) e. D )
27 10 eldifad 
 |-  ( ph -> s e. D )
28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 hdmaprnlem4tN 
 |-  ( ph -> t e. V )
29 1 2 3 5 15 8 9 28 hdmapcl 
 |-  ( ph -> ( S ` t ) e. D )
30 15 18 22 25 26 27 29 25 26 27 29 ablpnpcan 
 |-  ( ph -> ( ( ( S ` u ) .+b s ) ( -g ` C ) ( ( S ` u ) .+b ( S ` t ) ) ) = ( s ( -g ` C ) ( S ` t ) ) )
31 15 18 lmodvacl 
 |-  ( ( C e. LMod /\ ( S ` u ) e. D /\ s e. D ) -> ( ( S ` u ) .+b s ) e. D )
32 23 26 27 31 syl3anc 
 |-  ( ph -> ( ( S ` u ) .+b s ) e. D )
33 eqid 
 |-  ( LSubSp ` C ) = ( LSubSp ` C )
34 15 33 6 lspsncl 
 |-  ( ( C e. LMod /\ ( ( S ` u ) .+b s ) e. D ) -> ( L ` { ( ( S ` u ) .+b s ) } ) e. ( LSubSp ` C ) )
35 23 32 34 syl2anc 
 |-  ( ph -> ( L ` { ( ( S ` u ) .+b s ) } ) e. ( LSubSp ` C ) )
36 15 6 lspsnid 
 |-  ( ( C e. LMod /\ ( ( S ` u ) .+b s ) e. D ) -> ( ( S ` u ) .+b s ) e. ( L ` { ( ( S ` u ) .+b s ) } ) )
37 23 32 36 syl2anc 
 |-  ( ph -> ( ( S ` u ) .+b s ) e. ( L ` { ( ( S ` u ) .+b s ) } ) )
38 15 18 lmodvacl 
 |-  ( ( C e. LMod /\ ( S ` u ) e. D /\ ( S ` t ) e. D ) -> ( ( S ` u ) .+b ( S ` t ) ) e. D )
39 23 26 29 38 syl3anc 
 |-  ( ph -> ( ( S ` u ) .+b ( S ` t ) ) e. D )
40 15 6 lspsnid 
 |-  ( ( C e. LMod /\ ( ( S ` u ) .+b ( S ` t ) ) e. D ) -> ( ( S ` u ) .+b ( S ` t ) ) e. ( L ` { ( ( S ` u ) .+b ( S ` t ) ) } ) )
41 23 39 40 syl2anc 
 |-  ( ph -> ( ( S ` u ) .+b ( S ` t ) ) e. ( L ` { ( ( S ` u ) .+b ( S ` t ) ) } ) )
42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 hdmaprnlem6N 
 |-  ( ph -> ( L ` { ( ( S ` u ) .+b s ) } ) = ( L ` { ( ( S ` u ) .+b ( S ` t ) ) } ) )
43 41 42 eleqtrrd 
 |-  ( ph -> ( ( S ` u ) .+b ( S ` t ) ) e. ( L ` { ( ( S ` u ) .+b s ) } ) )
44 22 33 lssvsubcl 
 |-  ( ( ( C e. LMod /\ ( L ` { ( ( S ` u ) .+b s ) } ) e. ( LSubSp ` C ) ) /\ ( ( ( S ` u ) .+b s ) e. ( L ` { ( ( S ` u ) .+b s ) } ) /\ ( ( S ` u ) .+b ( S ` t ) ) e. ( L ` { ( ( S ` u ) .+b s ) } ) ) ) -> ( ( ( S ` u ) .+b s ) ( -g ` C ) ( ( S ` u ) .+b ( S ` t ) ) ) e. ( L ` { ( ( S ` u ) .+b s ) } ) )
45 23 35 37 43 44 syl22anc 
 |-  ( ph -> ( ( ( S ` u ) .+b s ) ( -g ` C ) ( ( S ` u ) .+b ( S ` t ) ) ) e. ( L ` { ( ( S ` u ) .+b s ) } ) )
46 30 45 eqeltrrd 
 |-  ( ph -> ( s ( -g ` C ) ( S ` t ) ) e. ( L ` { ( ( S ` u ) .+b s ) } ) )